Connection Laplacian

The connection Laplacian on a vector bundle with connection is . A connection is a covariant derivative, saying how to differentiate sections and, once integrated, how to transport a fibre element along a path. Its Dirichlet energy measures how far a section is from being parallel, or covariantly constant. The kernel is exactly the parallel sections.

The discrete case is a connection graph

A cellular-sheaf whose restriction maps are invertible, a local system, is the discrete analogue of a bundle with connection. Each edge carries an isomorphism transporting the stalk of to that of . Then with , the connection-graph form of Singer–Wu’s vector diffusion maps, generalising the graph Laplacian (all ) to the case where agreement is measured after transport.

Smoothness by another name

The connection Laplacian penalises disagreement between neighbours after parallel transport. Where the Hodge Laplacian of a plain graph penalises disagreement between neighbours directly, compares them only once transported, so its minimisers vary smoothly along the geometry rather than being locally constant. It reduces to the sheaf Laplacian when the connection is trivial, and to the graph Laplacian when both the connection and the stalks are.

References

• Singer & Wu, “Vector Diffusion Maps and the Connection Laplacian,” Communications on Pure and Applied Mathematics (2012)
• Hansen & Ghrist, “Toward a Spectral Theory of Cellular Sheaves,” Journal of Applied and Computational Topology (2019)